Math 216: Foundations of algebraic geometry 2009-10

An updated version of the notes will be gradually posted here, starting roughly September 1, 2010. See this mathoverflow page for interesting related discussion.

The table of contents (and sketchy introduction) for the notes as of June 15, 2010
(much of the course) is here.
Topics near the end are quite rough (see the bottom of this page for further discussion). I'll drop off copies in the mailboxes of many of the people in the class late in the week of June 15. A paper copy of the current version will be available outside my office then too.

There are several types of courses that can go under the name of
"introduction to algebraic geometry": complex geometry; the theory of
varieties; a non-rigorous examples-based course; algebraic geometry
for number theorists (perhaps focusing on elliptic curves); and more.
There is a place for each of these courses. This course will deal
with schemes, and will attempt to be faster and more complete and
rigorous than most, but with enough examples and calculations to help
develop intuition for the machinery.
Such a course is normally a
"second course" in algebraic geometry, and in an ideal world, people
would learn this material over many years. We do not live in an ideal
world. To make things worse, I am experimenting with the material,
and trying to see if a non-traditional presentation will make it possible
to help people learn this material better, so this
year's course is only an approximation.
(See
here
for an earlier version.)

This course is for mathematicians intending to get near the boundary of current research, in algebraic geometry or a related part of mathematics. It is not intended for undergraduates or people in other fields; for that, people should take Maryam Mirzakhani's class, or else wait for a later incarnation of Math 216 (which will vary in style over the years).

In short, this not a course to take casually. But
if you have the interest and time and energy, I will do my best to
make this rewarding.

Office hours:
Because of the nature of this class, I'd like to be as open as possible about office hours, and not have them restricted to a few hours per week. So if you would like to chat, please let me know, and I'll be most likely happy to meet on a couple of days' notice. I am almost always available to meet immediately after class.

References:
I hope to periodically release notes,
perhaps once per week.
(The most recent version should be outside my office door.
I will also give each new version out in class as they become available.)
You should take notes yourself, and not count on these.
The notes from the class two years ago are available
here.
It may be useful having Hartshorne's
Algebraic Geometry, and possibly Mumford's Red Book of Varieties and Schemes
(the original edition is better, as Springer introduced errors
into the second edition by retyping it). Mumford (second edition) is availble online
(with a Stanford account) from Springer.
For background on commutative algebra, I'd suggest consulting
Eisenbud's Commutative Algebra with a View toward Algebraic Geometry
or Atiyah and MacDonald's
Commutative Algebra. For background on abstract nonsense,
Weibel's Introduction to Homological Algebra is good to have handy.
Freyd's Abelian Categories
is available online (free and legally) here.

Homework: Unlike most advanced graduate courses, there
will be homework. It is important --- this material is very dense,
and the only way to understand it is to grapple with it at close range.
There will be a problem set most weeks.
Your grade will depend on the problem sets.

Fall quarter

Mon. Sept. 21: introduction. Why you shouldn't take this class. What I'm trying to do in this class. What algebraic geometry is about. That's too hard, so at least what this course is about: why many notions (geometric, arithmetic, algebraic, complex-analytic, ...) can be understood in terms of "geometric spaces", and constructions related to them. Example: Mordell's Conjecture (Faltings' Theorem). A little bit of category theroy: objects, morphisms, source, target, identity, isomorphism, automorphism, examples (sets, vector spaces, A-modules, abelian groups, rings, topological spaces, partially order sets, the open sets of a topological space, subcategory.

Wed. Oct. 7: base of topology, sheaf on a base, sheaves on a base are same as sheaves (including morphisms). Toward schemes: set, topology, sheaf of functions, and more motivation from manifolds. Affine schemes: Spec A as a set,
and a preliminary example. Functions, and values at a point.

No class Oct. 14 or 16. Instead, read to the end of the chapter 3 notes given out on Mon. Oct. 12. Words you should be comfortable with (in this setting): irreducible, closed point, quasicompact, specialization, generization, generic point, Noetherian topological space, Hilbert basis theorem, A[[x]], irreducible component, Noetherian induction, connected, conected component, the function I from subsets of Spec A to ideals of A.
Jack is happy to meet on Thursday (as well as his usual office hours);
e-mail him if you'd like to set up a time.

Mon. Oct. 19: the structure sheaf of Spec A (by defining the sections over D(f) as the localization of A at those functions nonvanishing in D(f), and showing this is a sheaf on the base); the O_{Spec A}-module M-tilde, isomorphism of ringed spaces, affine scheme, scheme, isomorphism of schemes, open subscheme, affine open subset/subscheme.

Wed. Oct. 28: associated points of schemes, and fuzzy mathematics; embedded point, rational function, domain of definition, regular at a point, , total fraction ring. Generic points are associated points; if X is reduced, then it has no embedded points; functions are determined by their germs at associated points; zero divisors are precisely those functions vanishing at an associated point of X Primary ideals, (minimal) primary decmposition (and its "uniqueness"), associated primes (of an ideal, of a ring).

Fri. Oct. 30: morphisms of schemes as maps of local-ringed spaces. Complex schemes, or more generally k-schemes (where k is a field), or more generaly A-schemes (A is a ring), or more generally S-schemes (S is a scheme).

Wed. Jan. 20: the local dimension is at most the dimension of the cotangent space, the
Jacobian criterion, Euler test, two facts stated but not proved (regular local rings are UFDs, and remain regular upon localization; won't be used). Seven faces of Discrete Valuation Rings.

Fri. Jan. 29: affine locality of quasicoherence. Pushforwards of quasicoherent sheaves by nice (quasicompact quasiseparated) morphisms are quasicoherent. Quasicoherent sheaves form an abelian category, and you can work affine-by-affine for all module-like constructions (tensor, Sym, wedge).

Fri. Feb. 19: quasicoherent sheaves on projectie A-schemes and projective modules. Getting a quasicoherent sheaf from a graded module (the "tilde" functor). O(n), and shifting of the index of a graded module. Globally generation, and Serre's theorem. Getting from quasicoherent sheaves to graded modules: the Gamma_* functor, which is adjoint to tilde. Every quasicoherent sheaf on Proj A arises form the tilde construction. Saturated modules.

Fri. Feb. 26: Revisiting the construction of pullback and relative Spec. Relative Proj, and projective morphisms. (Sample consequence: an integral curve over a field has a birational model that is nonsingular and projective.)

Mon. Mar. 8 (applications of cohomology to projective k-schemes): Euler characteristic (additive in exact sequences), Riemann-Roch for line bundles (or coherent sheaves) on a curve, degree of a line bundle or coherent sheaf on a curve, rank Hilbert function, Hilbert polynomial (of a coherent sheaf F has degree equal to the dimension of the support), Bezout's theorem (intersecting a hypersurface with any subvariety), arithmetic genus.

Mon. Apr. 12: the "Clear-denominators" extension theorem (from smooth curves to projective schemes); various "categories of curves" are all equivalent.

Wed. Apr. 14: degree of a morphism between nonsingular curves.
To set up a discussion of curves: discussion of differentials and Serre duality (e.g. the degree of the canonical bundle is 2g-2).

Mon. Apr. 19: the Riemann-Hurwitz formula, and a criterion for a morphism to be a closed immersion.

Wed. Apr. 21: a series of crucial observations in preparation for the study of curves. (Conclusion: if L has degree at least 2g, it gives a morphism to projective space; if it has degree greater than 2g, it gives a closed immersion.) Curves of genus 0. The trichotomy of curves (genus 0, 1, and greater than 1), and how it generalizes.

Mon. May 10: (counter)examples using elliptic curves (a factorial scheme with every affine open not UFD; a crazy Picard group; an affine open of an affine that is not distinguished; a variety with non-finitely-generated space of global sections; a proper nonprojective surface).

Wed. May 12: differentials.

Fri. May 14: differentials and the Riemann-Hurwitz formula.differentials conti.

Mon. May 31: smooth, etale, unramified. The only hard fact: the conormal exact sequence is exact on the left if X/Y is smooth.

Wed. June 2: proof of Serre duality for curves (and, with black boxes of miracle flatness and cup product, in higher dimension). The dualizing sheaf for curves is an invertible sheaf (a line bundle). Sketch of why it is the sheaf of differentials.

Thurs. June 3, at noon (in the Faculty Area Research Seminar Series):
plausibility argument that the moduli space of curves of genus g greater than 2 has dimension 3g-3.

There are a few topics we didn't get to. In the notes, I've
written some up, including a proof of cohomology and base change
theorem and related facts, a full proof of Serre duality (including
that the determinant of Omega is dualizing), Chow's Lemma (and the fact that coherent sheaves push forward to coherent sheaves under proper morphisms, with civilized hypotheses), and blowing up. There are some other
topics I intend to type up in the medium term (some by later this
summer), notably ampleness
and possibly Cohen-Macaulayness.
I would like to write up some notes on things related to Stein factorization (and other things related to when the pushforward of the structure sheaf is the structure sheaf) and Zariski's Main Theorem. But I'm not sure when I'll get to that.
I may
try to do some examples (surfaces, K3 surfaces and Calabi-Yau
varieties, abelian varieties, toric varieties). If there are other
things that you think I should do (even if you didn't take the class),
please let me know!
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